Mathematics Test 120

Result

Mathematics Test 120

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Weekly Quiz Competition

Question 1
4 / -1

Let α and β be the roots of the equation

x²– ax + b = 0 and A_n = αⁿ + βⁿ, then

A_n+1 – a.A_n + b. A_n–1 is equal to

A
–a

B
b

C
0

D
a - b

Solution

α + β = a, αβ = b

A_{n + 1} – a.A_n + b.A_{n – 1}

⇒ (α^{n + 1} + β^{n + 1}) – (α + β) (αⁿ + βⁿ) + αβ(β^{n – 1} + β^{n – 1})

⇒ α^{n + 1} + β^{n + 1} – α^{n + 1} – αβⁿ – βαⁿ – β^{n + 1} + αⁿβ + αβⁿ= 0
Question 2
4 / -1

If the quadratic equation ax² + bx + 6 = 0 does not have two distinct real roots, then least value of 2a + b is

A
2

B
-3

C
-4

D
1

Solution

ƒ(x) = ax² + bx + 6

its possible graph is

ƒ(2) = 4a + 2b + 6 ≥ 0

∴ 2a + b ≥ –3

and curve posses through (0,6)

∴ least value of 2a + b = –3
Question 3
4 / -1

Maximum value of f(x) = 4+6{x} / 1+2{x} is (where {.} denotes fractional part function)

A
1

B
2

C
3

D
4

Solution

f(x) = 3 + 1/1+2{x}

ƒ(x)_max. = 4
Question 4
4 / -1

If α, β are roots of equation 2x² + 7x + 3/2 = 0, then value of cot^–1α + cot^–1β is equal to -

A
π – tan^–1(5)

B
2π – tan^–1(7)

C
π + (tan^–12)

D
π + tan^–1(14)

Solution
Question 5
4 / -1

If ƒ(x) = sin^–1x + cos^–1x + tan^–1x + cot^–1x + sec^–1x, then which of the following statement(s) is/are INCORRECT ?

A
Domain of ƒ has infinitely many elements

B
Range of ƒ has infinitely many elements

C
Range of ƒ = [π, 2π]

D
All are correct

Solution

Domain = {–1, 1}

Range = {π, 2π}
Question 6
4 / -1

Let g(x) = x² – 2x + 2 and ƒ(x) = x² + x + α. If ƒ(g(x)) = 0 has no real solution then the set of values of a is a subset of -

A
(0,∞)

B
[–2,∞)

C
[–10,∞)

D
option 2 and option 3 are correct

Solution

g(x) = (x – 1)² + 1 ⇒ g(x) > 1

ƒ(g(x)) = (g(x))² + g(x) + α = 0
Question 7
4 / -1

A monic quadratic polynomial y = ax² + bx + c having roots of opposite sign then which of the following can be correct ?

A
- b/2a > 0

B
- b/2a < 0

C
ac – ab² < 0

D
All are correct

Solution

Here a = 1

ƒ(x) = y = x² + bx + c

Vertex : x = - b/2a

can take any real value

ƒ(0) = c < 0.
Question 8
4 / -1

If exactly one root of equation x² – (p + 1)x + p = 0 lies in (0,4), then value of p can be-

A
0

B
4

C
6

D
All are correct

Solution

roots x = p; x = 1

∵ 1 lies in (0,4)

∴ p ≥ 4 or p ≤ 0
Question 9
4 / -1

Let α, β & γ are roots of equation x³ – λx² – 4 = 0 such that (2 + α) (2 + β) (2 + γ) is equal to 8, then

α² + β² + γ² is equal to-

A
1

B
4

C
9

D
16

Solution

x³ – λx² – 4 = (x – α) (x – β) (x – γ)

Put x = –2 ⇒ –4λ – 12

= – (2 + α) (2 + β) (2 + γ)

⇒ λ = –1

⇒ α + β + γ = –1, αβ + βγ + γα = 0,αβγ = 4

⇒ α² + β² + γ² = 1 – 0 ⇒ 1
Question 10
4 / -1

Let α, β & γ are roots of equation x³ – λx² – 4 = 0 such that (2 + α) (2 + β) (2 + γ) is equal to 8, then

(4αβ – 1) (4βγ – 1) (4γα – 1) is equal to-

A
4

B
256

C
1804

D
1087

Solution

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Re-Attempt Weekly Quiz Competition

Mathematics Test 120

Result

Mathematics Test 120

Score

-

Rank

-

SHARING IS CARING

Question 1

–a

b

0

a - b

Solution

Question 2

2

-3

-4

1

Solution

Question 3

1

2

3

4

Solution

Question 4

π – tan–1(5)

2π – tan–1(7)

π + (tan–12)

π + tan–1(14)

Solution

Question 5

Domain of ƒ has infinitely many elements

Range of ƒ has infinitely many elements

Range of ƒ = [π, 2π]

All are correct

Solution

Question 6

(0,∞)

[–2,∞)

[–10,∞)

option 2 and option 3 are correct

Solution

Question 7

- b/2a > 0

- b/2a < 0

ac – ab2 < 0

All are correct

Solution

Question 8

0

4

6

All are correct

Solution

Question 9

1

4

9

16

Solution

Question 10

4

256

1804

1087

Solution

User

Question Analysis

My Perfomance

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-

Rank

-

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π – tan^–1(5)

2π – tan^–1(7)

π + (tan^–12)

π + tan^–1(14)

ac – ab² < 0